A cubic function is a polynomial of degree three.
e.g. y = x^{3} + 3x^{2} − 2x + 5
Cubic graphs can be drawn by finding the x and y intercepts.
Because cubic graphs do not have axes of symmetry the turning points have to be found using calculus.
Sketching Cubics
Method 1: Factorisation.
If the equation is in the form y = (x − a)(x − b)(x − c) the following method should be used:
Step 1: Find the xintercepts by putting y = 0. Step 2: Find the yintercept by putting x = 0. Step 3: Plot the points above to sketch the cubic curve.
e.g. Sketch the graph of y = (x − 2)(x + 3)(x − 1)
Step 1:
Find the xintercepts by putting y = 0.
0 = (x − 2)(x + 3)(x − 1)
x = 2 or 3 or 1
Step 2:
Find the yintercepts by putting x = 0.
y = (0 − 2)(0 + 3)(0 − 1)
y = 2 x 3 x 1
y = 6
Step 3:
Plot the points and sketch the curve.
Note: Functions with a repeated factor have a graph which just touches the xaxis. e.g. y = (x − 2)^{2}(x + 1)
Method 2: Transformation
The graph of the basic cubic y = x^{3} is shown in the diagram.
This basic cubic is moved or transformed as follows:
y = ax^{3} The a has the effect of changing the basic cubic in the y direction. It affects the steepness of the graph.


y = x^{3} + k The k has the effect of moving the cubic up or down the yaxis by k units.


y = (x − h)^{3} The h has the effect of moving the basic cubic along the xaxis by h units. 